49 research outputs found
Representing Primes as the Form in Some Imaginary Quadratic Fields
We give criteria of the solvability of the diophantine equation
over some imaginary quadratic fields where is a prime element. The criteria
becomes quite simple in special cases.Comment: 8 pages, This paper has been withdrawn by the author since it was
merged into the article arXiv:1405.5776 on August 8, 201
Convergence, Finiteness and Periodicity of Several New Algorithms of p-adic Continued Fractions
-adic continued fractions, as an extension of the classical concept of
classical continued fractions to the realm of -adic numbers, offering a
novel perspective on number representation and approximation. While numerous
-adic continued fraction expansion algorithms have been proposed by the
researchers, the establishment of several excellent properties, such as the
Lagrange Theorem for classic continued fractions, which indicates that every
quadratic irrationals can be expanded periodically, remains elusive. In this
paper, we present several new algorithms that can be viewed as refinements of
the existing -adic continued fraction algorithms. We give an upper bound of
the length of partial quotients when expanding rational numbers, and prove that
for small primes , our algorithm can generate periodic continued fraction
expansions for all quadratic irrationals. As confirmed through experimentation,
one of our algorithms can be viewed as the best -adic algorithm available to
date. Furthermore, we provide an approach to establish a -adic continued
fraction expansion algorithm that could generate periodic expansions for all
quadratic irrationals in for a given prime
Exact Covering Systems in Number Fields
It is well known that in an exact covering system in , the
biggest modulus must be repeated. Very recently, Kim gave an analogous result
for certain quadratic fields, and Kim also conjectured that it must hold in any
algebraic number field. In this paper, we prove Kim's conjecture. In other
words, we prove that exact covering systems in any algebraic number field must
have repeated moduli.Comment: 13 page